Optimized equations for X_{1}(N)
The table below gives links to optimized equations f(x,y)=0 for X_{1}(N), together with parametizations
E=[a_{1}(x,y),a_{2}(x,y),a_{3}(x,y),a_{4}(x,y),a_{6}(x,y)] and
P=[u(x,y),v(x,y)]
that define an elliptic curve
v^{2} + a_{1}uv + a_{3}v = u^{3} + a_{2}u^{2} + a_{4}u + a_{6}
in Weierstrass form on which P is a point of order N.
This extends results for N ≤ 50 described in Constructing elliptic curves with prescribed torsion over finite fields to N ≤ 100, and for many N ≤ 50 gives equations of lower degree (for N > 100, see this table).
In each case the equations below give a map to P^{1} (the function y) that matches the upper bound on the gonality of X_{1}(N) given by Derrickx and van Hoeij
in Gonality of the modular curve X_{1}(N). These bounds are known to be tight for N ≤ 40.
This is joint work in progress with Mark van Hoeij (last updated January 10, 2014). Partially funded by NSF grant DMS1115455.
N  deg_{x}  deg_{y}  f(x,y)
 N  deg_{x}  deg_{y}  f(x,y)
 N  deg_{x}  deg_{y}  f(x,y)
 N  deg_{x}  deg_{y}  f(x,y)
 N  deg_{x}  deg_{y}  f(x,y)

6  0  1  f(x,y)
 25  8  5  f(x,y)
 44  22  15  f(x,y)
 63  72  36  f(x,y)
 82  63  58  f(x,y)

7  0  1  f(x,y)
 26  7  6  f(x,y)
 45  36  18  f(x,y)
 64  48  32  f(x,y)
 83  95  90  f(x,y)

8  0  1  f(x,y)
 27  8  6  f(x,y)
 46  21  19  f(x,y)
 65  53  42  f(x,y)
 84  72  48  f(x,y)

9  0  1  f(x,y)
 28  9  6  f(x,y)
 47  30  29  f(x,y)
 66  60  30  f(x,y)
 85  90  72  f(x,y)

10  0  1  f(x,y)
 29  12  11  f(x,y)
 48  24  16  f(x,y)
 67  59  58  f(x,y)
 86  73  64  f(x,y)

11  2  2  f(x,y)
 30  8  6  f(x,y)
 49  31  21  f(x,y)
 68  54  36  f(x,y)
 87  140  70  f(x,y)

12  0  1  f(x,y)
 31  12  12  f(x,y)
 50  23  15  f(x,y)
 69  88  44  f(x,y)
 88  90  60  f(x,y)

13  3  2  f(x,y)
 32  12  8  f(x,y)
 51  48  24  f(x,y)
 70  45  36  f(x,y)
 89  108  104  f(x,y)

14  2  2  f(x,y)
 33  20  10  f(x,y)
 52  31  21  f(x,y)
 71  69  66  f(x,y)
 90  67  48  f(x,y)

15  2  2  f(x,y)
 34  10  10  f(x,y)
 53  39  37  f(x,y)
 72  45  32  f(x,y)
 91  106  84  f(x,y)

16  3  2  f(x,y)
 35  15  12  f(x,y)
 54  25  18  f(x,y)
 73  73  70  f(x,y)
 92  99  66  f(x,y)

17  4  4  f(x,y)
 36  11  8  f(x,y)
 55  37  30  f(x,y)
 74  52  51  f(x,y)
 93  160  80  f(x,y)

18  3  2  f(x,y)
 37  19  18  f(x,y)
 56  36  24  f(x,y)
 75  63  40  f(x,y)
 94  83  83  f(x,y)

19  5  5  f(x,y)
 38  13  12  f(x,y)
 57  60  30  f(x,y)
 76  67  45  f(x,y)
 95  113  90  f(x,y)

20  3  3  f(x,y)
 39  28  14  f(x,y)
 58  32  31  f(x,y)
 77  74  60  f(x,y)
 96  81  56  f(x,y)

21  5  4  f(x,y)
 40  18  12  f(x,y)
 59  48  46  f(x,y)
 78  84  42  f(x,y)
 97  129  123  f(x,y)

22  5  4  f(x,y)
 41  23  22  f(x,y)
 60  35  24  f(x,y)
 79  85  82  f(x,y)
 98  92  63  f(x,y)

23  7  7  f(x,y)
 42  24  12  f(x,y)
 61  51  49  f(x,y)
 80  72  48  f(x,y)
 99  180  90  f(x,y)

24  5  4  f(x,y)
 43  26  24  f(x,y)
 62  36  36  f(x,y)
 81  77  54  f(x,y)
 100  95  60  f(x,y)
